Answer :
To determine the domain of the function \( h(x) = \sqrt{-x + 3} \), we need to ensure that the expression inside the square root is non-negative. The square root function is only defined for non-negative values, so let's set up an inequality to find the values of \( x \) that make \(-x + 3 \) non-negative.
1. Start with the expression inside the square root:
[tex]\[ -x + 3 \][/tex]
2. Set up the inequality to ensure it is non-negative:
[tex]\[ -x + 3 \geq 0 \][/tex]
3. Solve the inequality step-by-step:
[tex]\[ -x + 3 \geq 0 \][/tex]
[tex]\[ 3 \geq x \][/tex]
[tex]\[ x \leq 3 \][/tex]
Thus, the inequality \( x \leq 3 \) describes all the values of \( x \) for which the function \( h(x) \) is defined. To express this in interval notation, we include all \( x \) from negative infinity up to and including 3:
[tex]\[ (-\infty, 3] \][/tex]
Therefore, the domain of the function [tex]\( h(x) = \sqrt{-x + 3} \)[/tex] is [tex]\(\boxed{(-\infty, 3]}\)[/tex].
1. Start with the expression inside the square root:
[tex]\[ -x + 3 \][/tex]
2. Set up the inequality to ensure it is non-negative:
[tex]\[ -x + 3 \geq 0 \][/tex]
3. Solve the inequality step-by-step:
[tex]\[ -x + 3 \geq 0 \][/tex]
[tex]\[ 3 \geq x \][/tex]
[tex]\[ x \leq 3 \][/tex]
Thus, the inequality \( x \leq 3 \) describes all the values of \( x \) for which the function \( h(x) \) is defined. To express this in interval notation, we include all \( x \) from negative infinity up to and including 3:
[tex]\[ (-\infty, 3] \][/tex]
Therefore, the domain of the function [tex]\( h(x) = \sqrt{-x + 3} \)[/tex] is [tex]\(\boxed{(-\infty, 3]}\)[/tex].
